The reigning champions found themselves 2-0 down within 15 minutes at the KC Stadium on Thursday, James Chester volleying home against his former club before David Meyler's strike found the net via a deflection from Jonny Evans.
United refused to wilt, however, and soon restored parity thanks to Chris Smalling's header and a superb volley from Wayne Rooney - his 150th goal in the Premier League for United.
In the second half, Bruce hit the crossbar when he met Tom Huddlestone's corner as Hull searched for a winner.
But it was United who managed to find the breakthrough, as Chester put past his own goalkeeper from Ashley Young's cross, as United made it 17 Premier League wins from their last18 Boxing Day fixtures.
Speaking after the game, Moyes praised his side's fighting qualities, while endorsing their title credentials.
"To be two down and come back to win is a great result," he told BBC Sport. "I thought it would be a long road back but these players have done this in the past and we did well to get back in the game.
"For us to get in at half-time 2-2 was great but when we were at 3-2 Hull gave us a few scary moments. However, I'd take that after being 2-0 down after 10 minutes.
"We're going to try to get the gap as narrow as we can. There's a long way to go in the league and we have to keep picking up points.
"There's a lot to do but we'll keep at it. I believe we can be in the mix and challenge whoever is near the top."
The victory was United's third win in a row in the Premier League, and Moyes' charges can make it four straight triumphs in the top flight in Saturday's visit to Norwich City.
